Math
Rørdam K-theory
2.2.10 Liftings
Chapter 2 Projections and Unitary elements
Chapter 3 The K_0 Group of a Unital C -algebra.
Chapter 4 The functor K0
Chapter 5 The ordered abelian group K0
Chapter 6 Inductive limits
Characterization for stable equivalence
Corollary 2.1.4
Definition 2.1.2.
Definition 2.2.1
Definition 2.3.1 (The semigroup of projections of A)
Definition 2.3.3. (The semigroup of projections)
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Definition 3.1.6 (Stable equivalence)
Definition 3.2.5 (Homotopy equivalence)
Definition 4.1.1 (K0, non-unital case)
Definition 4.2.1 (The Scalar mapping)
Definition 5.1.1.
Definition 5.1.3
Definition 5.1.4 (Positive cone of K0)
Definition 5.16
Definition 6.4.a (Stabilization)
Drawing 2024-02-05 22.07.46
DRAWING
Example 3.1.3
Example 3.3.2
Example 3.3.3
Example 3.3.5
Example 4.1.5
Example 4.3.5
Lemma 2.1.3
Lemma 2.1.5 (Whitehead)
Lemma 2.1.7.
Lemma 2.2.3
Lemma 3.2.7
Lemma 3.2.8
Lemma 4.3.2
Lemma 5.1.2
Lemma 6.3.1
Murray-von Neumann equivalence
Paragraph 3.3.1 Traces and K0
Properties of the Grothendieck group
Proposition 2.1.6.
Proposition 2.1.8
Proposition 2.1.11
Proposition 2.2.5
Proposition 2.2.6
Proposition 2.2.7
Proposition 2.2.8
Proposition 2.3.2.
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Proposition 3.1.8 (Universal property of K0)
Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
Proposition 3.2.6 (Homotopy invariance of K0)
Proposition 4.1.3 (Functoriality of K0)
Proposition 4.1.4 (Homotopy invariance of K0)
Proposition 4.3.2 (Half exactness of K0)
Proposition 4.3.3 (Split exactness of K0)
Proposition 4.3.4 (Direct sums)
Proposition 4.3.8 (Stability)
Proposition 5.1.5
Proposition 5.1.7
Proposition 5.1.9.
Proposition 6.2.4
Proposition 6.4.1 (Stability of K0)
retract
Rørdam K-theory
The functor K00 3.2.3
The functor K0 for unital C-star Algebras 3.2.2
The Grothendieck construction 3.1.1
The group K_00 3.1.5
Theorem 2.1.9. (The polar decomposition)
Theorem 2.1.10 (Carl Neumann Series)
Theorem 4.1.2. (The functor K0)
Theorem 5.2.1
Theorem 5.2.2
Theorem 6.3.2 (Continuity of K0)
Diffuse measure
dirac measures
finite C-star algebra
finite projection
homotopy
infinite C-star algebra
infinite projection
probability measure
quasi-trace
stably finite

Lemma 2.1.3

Let be a unital C-star-Algebra, then

  1. For each self-adjoint element ,
  2. If is a unitary element in with , then .
  3. If and are unitary elements in with then

Proof of 1:
It follows from the 2.1.13 Theorem (Continuous functional calculus) That if is continuous and if is self-adjoint in , then This is only possible because on inverse and conjugate coincide.
This implies that is a unitary, and in particular is a unitary element in .
For each define via .

Since is continuous, so is the path in .
Since the map is continuous for t at every x, in particular it should be continuous at just h
Therefore .

Proof of 2:
If , then does not belong to for some real number . Let be the real function on the spectrum of defined by , where .
Then is continuous, and for every .
Set . It follows that is a self-adjoint element in with , and so belongs to by part 1.
The main idea here is that if the element has as spectrum only part of the unit circle, we can define a branch on the argument function acting on the spectrum of the element, from there we can access our element as an argument of some self adjoint element on the unit circle, then we apply part 1 to this exponential.

Proof of 3:
If , then because every unitary is of norm 1. This implies that . (Spectral radius and norm are equal for self-adjoints)
This in turn implies that . Now we summon part 2 to conclude that and finally that as desired.
This proof exploits the relationship between the spectral radius of a normal element and its norm, along with the convenient fact that
$v^{*}u\sim{h}1$ and $v\sim{h}v\implies v(v^{*}u)\sim_{h}v$ .

Table Of Contents
Lemma 2.1.3